Question

We are given a stick that extends from 0 to x. Its length, x, is the realization of an exponential random variable X, with mean 1. We break that stick at a point Y that is uniformly distributed over the interval [0,x].

1. Write down the (fully specified) joint PDF fX,Y(x,y) of X and Y.

   For 0<y≤x:

   fX,Y(x,y)=

2. Find Var(E[Y∣X]).

   Var(E[Y∣X])=

3. We do not observe the value of X, but are told that Y=2.2. Find the MAP estimate of X based on Y=2.2.

   MAP estimate of X:

Answer 1

Answer:-

Given That:-

We are given a stick that extends from 0 to x. Its length, x, is the realization of an exponential random variable X, with mean 1. We break that stick at a point Y that is uniformly distributed over the interval [0,x].

Write down the (fully specified) joint PDF fX,Y(x,y) of X and Y. For 0<y≤x:

Given That

fX,Y(x,y)=

Therefore Joint pdf of (X, Y) is

Find Var(E[Y∣X]).

Var(E[Y∣X])= Var(X/2) = 1/4 Var(X)

= 2

We do not observe the value of X, but are told that Y=2.2. Find the MAP estimate of X based on Y=2.2.

MAP estimate of X:

Poission distribution of X and Y is

k is such that

Given That y = 2.2 & thus we calculate

(using technology)

Therefore k = 1/.0371911 = 26.88815

= MAP estimator of x

= value of x maxizing for given y = 2.2

For y = 2.2

2.2 < x <

is maximum at x = 2.2

= MAP estimator of x for y = 2.2

  

= 1.354224

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